Find Length of r(t) on [0,3]: Sketching the Plane Curve in xy-Plane

In summary, the given plane curve can be represented as a straight line on the xy-plane. Its length over the given interval can be found by calculating the integral of the magnitude of its derivative, which is equal to 10. To sketch the curve, find the points corresponding to t=0 and t=3 and draw a straight line between them.
  • #1
p4nda
16
0
Sketch the plane curve in the xy-plane and find its length over the given interval:
r(t) = (6t-3)i + (8t+1)j on [0,3]

Here's what I've got so far:
r'(t) = 6i + 8j
llr'(t)ll = sqrt of 6^2+8^2 = 10
s = integral 0-3 10dt
= 10x ]0 to 3
= [30-0]
= 30I just need help on how to sketch this plane curve. Thanks.
 
Last edited:
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  • #2
to sketch curves, you can only find the derivative to get an idea of how the curve behave and then just calculate a bunch of points and link them together. it's tedious and boring, just hang on.
 
  • #3
Since the functions for the x and y components are linear, this is, of course, a straight line! Find the point corresponding to t= 0, the point correponding to t= 3 and draw the straight line between them.
 

FAQ: Find Length of r(t) on [0,3]: Sketching the Plane Curve in xy-Plane

1. How do I find the length of a plane curve?

To find the length of a plane curve, you need to use the arc length formula, which is given by:

L = ∫ab √(1 + [r'(t)]2) dt

where r(t) is the parametric equation of the curve and a and b are the limits of integration.

2. What is the significance of the interval [0,3] in finding the length of r(t)?

The interval [0,3] represents the range of values for t, which is the independent variable in the parametric equation of the curve. This interval determines the portion of the curve that will be used in the length calculation.

3. How do I sketch a plane curve in the xy-plane?

To sketch a plane curve in the xy-plane, you need to plot points on a Cartesian coordinate system using the parametric equation of the curve. The x values will be given by rx(t) and the y values will be given by ry(t). You can then connect the points to create a smooth curve.

4. What is the difference between a plane curve and a regular curve?

A plane curve is a curve that lies entirely on a plane, such as the xy-plane. A regular curve, on the other hand, can be any curve in three-dimensional space. Plane curves are often easier to work with mathematically because they have fewer variables.

5. Can I use the arc length formula to find the length of any curve?

No, the arc length formula can only be used to find the length of a curve that is described by a parametric equation. If the curve is described by a function in the form of y = f(x), you will need to use a different formula to find its length.

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